find \\( \\frac { d r } { d \\theta } \\).\n\n\\( r = 1 - \\theta ^ { 4 } \\sin \\theta \\)\n\n\\( \\frac {…

Explanation:

Step1: Differentiate the constant term

The derivative of a constant (1) with respect to (\theta) is (0). So, (\frac{d}{d\theta}(1) = 0).

Step2: Apply the product rule

The product rule states that if (u = \theta^{4}) and (v=\sin\theta), then (\frac{d}{d\theta}(uv)=u'v + uv').

• Find (u'): (\frac{d}{d\theta}(\theta^{4})=4\theta^{3}) (using the power rule (\frac{d}{d\theta}(\theta^{n})=n\theta^{n - 1})).
• Find (v'): (\frac{d}{d\theta}(\sin\theta)=\cos\theta).
• Then (\frac{d}{d\theta}(\theta^{4}\sin\theta)=4\theta^{3}\sin\theta+\theta^{4}\cos\theta).

Step3: Combine the results

Since (r = 1-\theta^{4}\sin\theta), (\frac{dr}{d\theta}=\frac{d}{d\theta}(1)-\frac{d}{d\theta}(\theta^{4}\sin\theta)). Substitute the values from Step1 and Step2: (\frac{dr}{d\theta}=0-(4\theta^{3}\sin\theta+\theta^{4}\cos\theta)=-\theta^{3}(4\sin\theta+\theta\cos\theta)).

Answer:

(-\theta^{3}(4\sin\theta+\theta\cos\theta))